In which of the following reaction, the value of `K_(p)` will be equal to `K_(c)` ?

Options

A. `H_(2)(g)+I_(2)(g) hArr 2HI (g)`
B. `PCl_(5)(g) hArr PCl_(3)(g)+Cl_(2)(g)`
C. `2NH_(3)(g) hArr N_(2)(g)+3H_(2)(g)`
D. `2SO_(2)(g)+O_(2)(g) hArr 2SO_(3)(g)`

Answer

To determine in which of the given reactions the value of \( K_p \) will be equal to \( K_c \), we need to use the relationship between \( K_p \) and \( K_c \):

\[ K_p = K_c R T^{\Delta n_g} \]

Where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( \Delta n_g \) is the change in the number of moles of gas, calculated as:

\[ \Delta n_g = \text{(sum of stoichiometric coefficients of products)} - \text{(sum of stoichiometric coefficients of reactants)} \]

For \( K_p \) to be equal to \( K_c \), \( \Delta n_g \) must be equal to 0. This means that the number of moles of gaseous products must equal the number of moles of gaseous reactants.

Now, let's analyze each option provided in the question:

### Option A:
\[ \text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2 \text{HI}(g) \]
- Stoichiometric coefficients:
- Products: 2 (from 2 HI)
- Reactants: 1 (from H2) + 1 (from I2) = 2
- Calculation of \( \Delta n_g \):
\[ \Delta n_g = 2 - 2 = 0 \]
Since \( \Delta n_g = 0 \), \( K_p = K_c \).

### Option B:
\[ \text{PCl}_5(g) \rightleftharpoons \text{PCl}_3(g) + \text{Cl}_2(g) \]
- Stoichiometric coefficients:
- Products: 1 (from PCl3) + 1 (from Cl2) = 2
- Reactants: 1 (from PCl5)
- Calculation of \( \Delta n_g \):
\[ \Delta n_g = 2 - 1 = 1 \]
Since \( \Delta n_g \neq 0 \), \( K_p \neq K_c \).

### Option C:
\[ 2 \text{NH}_3(g) \rightleftharpoons \text{N}_2(g) + 3 \text{H}_2(g) \]
- Stoichiometric coefficients:
- Products: 1 (from N2) + 3 (from H2) = 4
- Reactants: 2 (from NH3)
- Calculation of \( \Delta n_g \):
\[ \Delta n_g = 4 - 2 = 2 \]
Since \( \Delta n_g \neq 0 \), \( K_p \neq K_c \).

### Option D:
\[ 2 \text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2 \text{SO}_3(g) \]
- Stoichiometric coefficients:
- Products: 2 (from SO3)
- Reactants: 2 (from SO2) + 1 (from O2) = 3
- Calculation of \( \Delta n_g \):
\[ \Delta n_g = 2 - 3 = -1 \]
Since \( \Delta n_g \neq 0 \), \( K_p \neq K_c \).

### Conclusion:
The only reaction for which \( K_p = K_c \) is in **Option A**.

### Final Answer:
**The value of \( K_p \) will be equal to \( K_c \) in Option A: \( \text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2 \text{HI}(g) \)**.

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